Geodesic
A complete complex hypersurface in the ball of $\mathbb{C}^N$
Josip Globevnik1
1 Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Annals of mathematics, Tome 182 (2015) no. 3, pp. 1067-1091.

Voir la notice de l'article provenant de la source Annals of Mathematics website

In 1977, P. Yang asked whether there exist complete immersed complex submanifolds $\varphi \colon M^k\rightarrow \mathbb{C}^N$ with bounded image. A positive answer is known for holomorphic curves $(k=1)$ and partial answers are known for the case when $k>1$. The principal result of the present paper is a construction of a holomorphic function on the open unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ whose real part is unbounded on every path in $\mathbb{B}_N$ of finite length that ends on $b\mathbb{B}_N$. A consequence is the existence of a complete, closed complex hypersurface in $\mathbb{B}_N$. This gives a positive answer to Yang’s question in all dimensions $k, N, 1\leq k, by providing properly embedded complete complex manifolds.
DOI : 10.4007/annals.2015.182.3.4

Josip Globevnik 1

1 Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia 
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Josip Globevnik. A complete complex hypersurface in the ball of $\mathbb{C}^N$. Annals of mathematics, Tome 182 (2015) no. 3, pp. 1067-1091. doi : 10.4007/annals.2015.182.3.4. http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.182.3.4/

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